CN113050683A - Fixed-time four-rotor aircraft control method based on terminal sliding mode control - Google Patents
Fixed-time four-rotor aircraft control method based on terminal sliding mode control Download PDFInfo
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Abstract
The invention discloses a fixed-time quadrotor aircraft control method based on terminal sliding mode control, which comprises the steps of establishing a quadrotor aircraft nonlinear dynamics model based on a Lagrange equation; performing control-oriented processing on the four-rotor model, and decoupling the four rotors into a position system and an attitude system based on a time scale decomposition method; a nonsingular terminal sliding mode strategy is adopted, a fixed time controller is designed for a position system and an attitude system, and the position error and the attitude error of the four-rotor system can tend to zero in fixed time; designing a nonsingular terminal sliding mode function based on a fixed time theory, wherein the upper bound of convergence time depends on the parameters of a controller and is irrelevant to the initial system state; simulation results prove that the terminal sliding mode fixed time controller designed by the invention has better convergence speed, avoids the problem of singularity, develops a good idea for the research of the control problem related to four rotors, and has the characteristics of good tracking capability, rapidity and robustness.
Description
Technical Field
The invention relates to the technical field of aircraft control, in particular to a fixed-time four-rotor aircraft control method based on terminal sliding mode control.
Background
The quad-rotor unmanned aerial vehicle has the flight advantages of small volume, light weight, fast flight and the like, has special performances of flexible flight attitude, free hovering and the like, is widely concerned, becomes a hotspot of directional research of unmanned aerial vehicles, and has a great deal of application in the fields of military, civil use and commerce; in military affairs, the system can not only detect, monitor and evaluate the enemy situation, but also be used for special tasks such as target search, communication relay, border patrol and the like, and even can be used as a miniature attack weapon in war to implement electronic warfare or directly attack targets on the other party; for civil use, unmanned aerial vehicles have been widely used in the fields of weather, communication, disaster monitoring and the like, for example, for Senda fire monitoring, searching disaster survivors or harmful gas pollution sources, volcanic exploration and other harsh environments that humans cannot reach; at present, the system is developed and applied to more industries such as civil aerial photography, cargo transportation, medical first aid, fault diagnosis and the like;
however, the control of the drone is very complicated by the high nonlinearity of the control system of the quad-rotor aircraft, the strong coupling between input and output variables, the uncertainty of the system itself and the unknown interference from the outside; the existing four-rotor aircraft control method mainly comprises the classical PID control method, the sliding mode variable structure control method, the fuzzy control method, the neural network control method and the like, and the methods have respective characteristics; how to design a control system which can be accurately controlled and has good stability is an important challenge for the development of a four-rotor aircraft and a main difficult problem to be solved;
in recent years, sliding mode variable structure control has the advantage of being unaffected by system parameters and external interference, and therefore is widely applied to the problem of nonlinear system control; the sliding mode variable structure control is a nonlinear algorithm with a variable control structure, and the working essence of the sliding mode variable structure control is that the control input of a controlled system is continuously adjusted according to state variables such as system state, deviation and the like, so that a system state track can reach and move along a predetermined sliding mode under the action of the control input until reaching a balance position; the method does not depend on an accurate model of the system, and the control effect is not easily influenced by mathematical parameters of the controlled object and various disturbance factors, so that the robustness is stronger compared with other control algorithms, and a good idea is developed for the research of the four-rotor related control problem.
Disclosure of Invention
Aiming at the existing problems, the invention aims to provide a fixed-time four-rotor aircraft control method based on terminal sliding mode control, and the nonsingular terminal sliding mode controller is designed by adopting a fixed-time theory, so that the obtained four-rotor aircraft has better tracking performance and certain rapidity and robustness, a good idea is developed for the research of the four-rotor related control problem, and the four-rotor aircraft control method has the characteristics of good tracking capability, rapidity and robustness.
In order to achieve the purpose, the technical scheme adopted by the invention is as follows:
a fixed-time four-rotor aircraft control method based on terminal sliding mode control comprises the steps of
The method comprises the following steps: firstly, establishing a nonlinear dynamics model of a quadrotor aircraft based on a Lagrange equation;
step two: converting a nonlinear dynamics model of the four-rotor aircraft into a second-order nonlinear system form comprising a position system and an attitude system;
step three: intermediate command signal theta for solving position model based on under-actuated characteristic of four-rotor aircraftd,ψd；
Step four: designing a nonsingular terminal sliding mode function based on a fixed time theory according to the model processing results of the second step and the third step;
step five: and designing a nonsingular terminal sliding mode fixed time controller by taking the nonsingular terminal sliding mode function in the step four as a control strategy, so that the tracking error of the position and posture trajectory of the system is converged to zero in fixed time.
Preferably, the nonlinear dynamical model of the quadrotor aircraft based on the lagrange equation in the step one is as follows:
wherein: the euler angles for the three attitudes of the aircraft are denoted as Ω ═ Φ, θ, ψ]Respectively representing a roll angle, a pitch angle and a yaw angle; angular velocity is expressed as
The position coordinate of the center of mass of the aircraft in the inertial coordinate system is represented as P ═ x, y, z](ii) a The velocity is expressed as
Aircraft radius length l represents the distance from each rotor tip to the aircraft center of gravity; m represents the total load weight of the four-rotor aircraft; i isiRepresenting the moment of inertia about each axis; kiIs a coefficient of resistance; di(i ═ 1,2,3,4,5,6) as perturbations, and a time-varying perturbation d is assumedi1.. 6 is bounded and known in the upper bound, i.e. there is a positive real number λ, such that | diThe | is less than or equal to lambda, and all disturbances are bounded;
using F as the thrust generated by each rotor of the aircraftiDenotes uiFor virtual control input, i ═ 1,2,3,4, …, defined as follows:
wherein: r represents a proportionality coefficient.
Preferably, the process of converting the nonlinear dynamical model of the quadrotor aircraft into the second-order nonlinear system form in the second step specifically includes:
s201, firstly, setting the virtual control input to be designed as follows:
the four-rotor dynamics model used to describe the position states in equation (1) becomes:
s202, order up＝[u1x,u1y,u1z]T，fp＝-[0,0,g]T-diag([K1/m,K2/m,K3/m]) V, let P ═ x, y, z]Representing three-dimensional position and v velocity, the quad-rotor aircraft position model in equation (1) can be written in the form of a second-order nonlinear system as follows:
s203. in the same way, make uο＝[u2,u3,u4]T，fο＝-diag[lK4/I1,lK5/I2,lK6/I3]·ω，dο＝[d4,d5,d6]TThen, the quad-rotor aircraft attitude model of equation (4) can be written in the form of a second-order nonlinear system as follows:
s204. set
The quad-rotor aircraft system can be converted to the following second-order system form:
preferably, the solving process for solving the intermediate command signal based on the under-actuated characteristic of the quadrotor aircraft in the third step includes:
s301. available from control inputs of a quad-rotor aircraft:
s302. because
Equation (8) becomes:
s303. due to u1z＝u1 cosφcosψdIs obtained by
Equation (9) becomes:
s304, the following formula (10) can be obtained:
s305. at this time, psi can be solved according to the formula (11)dAnd thetadComprises the following steps:
preferably, θ in step S305dThe virtual reference instruction is
Wherein:
preferably, the specific process for designing the nonsingular terminal sliding-mode function based on the fixed time theory in the step four includes:
s401, aiming at a second-order nonlinear system
If x is 0, the equilibrium state of the system is defined, and if there is a continuous radially unbounded function V: r → R+U {0}, so that
And the arbitrary solution x (t) of the system satisfies the formula
In formula (14): a. b, p, q and k are positive numbers and satisfy pk < 1 and qk > 1, then the zero balance state of the system is globally fixed time stable, and the solution time upper limit T satisfies the following inequality:
s402, setting a tracking error
Constructing the following nonsingular terminal sliding mode surfaces according to a fixed time theory:
in the formula (16), ai＞0，bi＞0，
Are all positive odd numbers, j is 1,2,3, …, and have
Preferably, the design process of the nonsingular terminal sliding mode fixed time controller in the step five includes:
in formula (17), k is 2 λ, and the nonlinear function μiThe definition is as follows:
in the formula (18), when x → 0, the nonlinear function μi(x)/x→0。
The invention has the beneficial effects that: the invention discloses a fixed-time four-rotor aircraft control method based on terminal sliding mode control, and compared with the prior art, the invention has the following improvement:
(1) aiming at the problems in the prior art, the invention designs a fixed-time four-rotor aircraft control method based on terminal sliding mode controld,yd,zd) And roll angle phidSolving the intermediate command signal theta on the basis of the position modeld,ψdAnd the attitude model is transmitted to finish the global control of the attitude and the position of the four-rotor aircraft;
(2) meanwhile, the control method constructs a nonsingular terminal sliding mode function based on a fixed time theory and provides an upper bound of system state convergence time irrelevant to an initial state; and by using a non-linear function mui(x) The strange problem is avoided; simulation results show that the controller designed by the control method has good robustness, can well execute a task of tracking a three-dimensional space track, and the convergence speed of system state errors is smaller than a given time upper bound, so that the effectiveness of the design is verified, and the method has the advantages of good tracking capability, rapidity and robustness.
Drawings
Fig. 1 is a control flow chart of a fixed-time four-rotor aircraft control method based on terminal sliding mode control according to the invention.
Fig. 2 is a 3D effect diagram of aircraft trajectory tracking in embodiment 1 of the present invention.
FIG. 3 is a graph of aircraft position tracking according to embodiment 1 of the present invention.
FIG. 4 is a graph of aircraft attitude tracking according to embodiment 1 of the present invention.
Fig. 5 is a position tracking error diagram according to embodiment 1 of the present invention.
Fig. 6 is a diagram of attitude tracking errors in embodiment 1 of the present invention.
Fig. 7 is a control input diagram of the position system in embodiment 1 of the present invention.
Fig. 8 is a control input diagram of the attitude system according to embodiment 1 of the present invention.
Detailed Description
In order to make those skilled in the art better understand the technical solution of the present invention, the following further describes the technical solution of the present invention with reference to the drawings and the embodiments.
Referring to the accompanying fig. 1-8, a fixed-time four-rotor aircraft control method based on terminal sliding mode control is provided, and the method adopts a fixed-time theory to design a nonsingular terminal sliding mode controller, and is performed according to the following steps:
the method comprises the following steps: firstly, establishing a nonlinear dynamics model of the quadrotor aircraft based on a Lagrange equation:
wherein: the euler angles for the three attitudes of the aircraft are denoted as Ω ═ Φ, θ, ψ]Respectively representing a roll angle, a pitch angle and a yaw angle; angular velocity is expressed as
The position coordinate of the center of mass of the aircraft in the inertial coordinate system is represented as P ═ x, y, z](ii) a The velocity is expressed as
Aircraft radius length l represents the distance from each rotor tip to the aircraft center of gravity; m represents the total load weight of the four-rotor aircraft; i isiRepresenting the moment of inertia about each axis; kiIs a coefficient of resistance; di(i ═ 1,2,3,4,5,6) as perturbations, and a time-varying perturbation d is assumedi1.. 6 is bounded and known in the upper bound, i.e. there is a positive real number λ, such that | diThe | is less than or equal to lambda, and all disturbances are bounded;
using F as the thrust generated by each rotor of the aircraftiDenotes uiFor virtual control input, i ═ 1,2,3,4, …, defined as follows:
wherein: r represents a proportionality coefficient;
step two: converting a nonlinear dynamics model of a four-rotor aircraft into a second-order nonlinear system form, specifically comprising
S201, firstly, in order to simplify the subsequent control algorithm analysis steps, the virtual control input to be designed is set as follows:
the four-rotor dynamics model used to describe the position states in equation (1) becomes:
s202, order up＝[u1x,u1y,u1z]T，fp＝-[0,0,g]T-diag([K1/m,K2/m,K3/m]) V, let P ═ x, y, z]Representing three-dimensional position and v velocity, the quad-rotor aircraft position model in equation (1) can be written in the form of a second-order nonlinear system as follows:
s203. in the same way, make uο＝[u2,u3,u4]T，fο＝-diag[lK4/I1,lK5/I2,lK6/I3]·ω，dο＝[d4,d5,d6]TThen, the quad-rotor aircraft attitude model of equation (4) can be written in the form of a second-order nonlinear system as follows:
s204, define
By integrating the above equations (2) to (6), the quad-rotor aircraft system can be converted into the following second-order system form:
step three: intermediate command signal theta for solving position model based on under-actuated characteristic of four-rotor aircraftd,ψdSpecifically, the tracking under the system under-actuation is four degrees of freedom, namely three-dimensional positions [ x, y, z ] respectively]And a roll angle phi, and the other two angles are ensured to be stable;
s301. available from control inputs of a quad-rotor aircraft:
s302. because
Equation (8) becomes:
s303. due to u1z＝u1 cosφcosψdIs obtained by
Equation (9) becomes:
s304, the following formula (10) can be obtained:
s305. at this time, psi can be solved according to the formula (11)dAnd thetadComprises the following steps:
if sin θ in formula (13)dBeyond [ -1,1 [ ]]Will cause theta todDoes not exist, i.e. cannot be solved, the solution is to thetadDesigning a virtual reference instruction as follows:
wherein:
step four: model processing in the second step and the third step enables the four-rotor model to meet a system form required by terminal sliding mode control, a nonsingular terminal sliding mode function based on a fixed time theory is designed, and the specific process is as follows:
s401, aiming at a second-order nonlinear system
If x is 0, the equilibrium state of the system is defined, and if there is a continuous radially unbounded function V: r → R+U {0}, so that
And the arbitrary solution x (t) of the system satisfies the formula
In formula (14): a. b, p, q and k are positive numbers and satisfy pk < 1 and qk > 1, then the zero balance state of the system is globally fixed time stable, and the solution time upper limit T satisfies the following inequality:
s403. setting the tracking error
Constructing the following nonsingular terminal sliding mode surfaces according to a fixed time theory:
in the formula (16), ai＞0，bi＞0，
Is an odd number, j is 1,2,3, …, and has
Step five: designing a nonsingular terminal sliding mode fixed time controller to make the tracking error of the system position and the attitude track converge to zero in fixed time, wherein the design process of the nonsingular terminal sliding mode fixed time controller comprises the following steps:
in formula (17), k is 2 λ, and the nonlinear function μiThe definition is as follows:
in the formula (18), when x → 0, the nonlinear function μi(x) The characteristic can ensure that the formula (18) in the controller is bounded, the occurrence of singular problems in the conventional sliding mode control is avoided, and under the action of the controller, the tracking error of the system position and posture track converges to zero at a fixed time,the procedure was demonstrated as follows:
according to equation (16), the differential of the sliding mode function can be obtained
By substituting formula (7) for formula (19), a compound of formula (I) can be obtained
The controller formula (17) is substituted into the formula (20) to obtain
Selecting a Lyapunov function Vi＝|si1., 6, from which the differential can be taken:
if it is not
From the formula (18)
Thus, it is possible to provide
Obtainable according to (14) and (15), ViWill be at a fixed time
Inner convergence to zero or into a region
Wherein
When in use
Based on formula (17), it is possible to obtain:
when in use
When there is
Can obtain ViWill be at a fixed time
Internal exit
Wherein:
thus, the convergence time upper bound can be expressed as
Under the action of the controller, the tracking errors of the position and attitude tracks of the four-rotor aircraft are converged to zero at fixed time, and the convergence time upper bound T is obtained under any initial conditionmaxIs determined by a control parameter ai＞0，bi＞0，
And τ, k.
Preferably, as a nonlinear control method, in step two, the six degrees of freedom, namely the position and the attitude, in the four-rotor aircraft dynamics model are all converted into corresponding six second-order nonlinear system modes (the four rotors are decoupled into a position system and an attitude system based on a time scale decomposition method), and the model is decoupled to meet the requirement of sliding mode variable structure control.
Preferably, as a global control strategy including attitude and position, in step three, the four-rotor aircraft has four inputs corresponding to the lift generated by the four rotors, respectively, and the outputs to be tracked are six, namely three-dimensional position and pitch, yaw and roll angles, which means that the four-rotor aircraft system has fewer inputs than outputs, and is a typical under-actuated system, and therefore, it is impossible to track six degrees of freedom simultaneously; one reasonable control scheme is to track the flight path (x)d,yd,zd) And roll angle phidIntermediate command signal thetad,ψdThe solution is needed according to the position model and is transmitted to the attitude model, and the overall control of the attitude and the position is completed.
Preferably, as a fixed time sliding mode control method, in the fourth step, an exponential-form nonsingular terminal sliding mode function is selected, so that the finally constructed controller meets the requirement that the system state tracking error is 0 within fixed time; in the traditional nonsingular sliding mode control method, a system reaches a sliding mode surface within limited time, but the convergence speed cannot be controlled, and the convergence time is influenced by an initial state; the nonsingular terminal sliding mode function is constructed according to a fixed time theory, the system state can be tracked, the time upper bound of 0 changed by the state tracking error can be calculated, the time upper bound is only related to design parameters and is unrelated to the initial state, and the timeliness and the reliability of system control are improved.
Preferably, as a control strategy, in step five, the controller design is carried out by adopting the nonsingular terminal sliding mode function based on the fixed time theory structure proposed in step four, and the nonlinear characteristic is added into the controllerFunction mui(x) And the occurrence of singular problems is avoided.
Example 1: step six, simulation experiments specifically comprise:
s601, establishing a four-rotor aircraft dynamic model by using a simulink module in an MATLAB simulation environment, wherein the four-rotor aircraft has the set parameters shown in a table 1:
table 1: four-rotor aircraft parameter setting
S602, designing a controller in an MATLAB simulation environment, wherein the controller parameters are as follows: value of external disturbance di0.2sin (i · t), i ═ 1,. 6; the surface coefficient of the position sliding mode is selected as
ai＝5,b i2, wherein i is 1,2, 3; the surface coefficient of the posture sliding mode is selected as
ai＝15,b i10, wherein i is 4,5, 6; controlling the gain k to be 1 and the time constant tau to be 0.1;
s603, setting the initial positions of the four rotors to be x, y and z]T＝[0,0,0]T(m) initial attitude angles of [ theta, psi, phi]T＝[0,0,0]T(rad); expected position instruction set to pd＝[0.5cos(0.5t),0.5sin(0.5t),0.1t]TThe desired roll angle is selectedd＝π/3；
S604, calculating an upper bound of convergence time according to set parameters, wherein the upper bound of the attitude convergence time is Tin3.1944s, upper bound T on the convergence time of the position loopout＝8.0255s。
An actual track and a reference track are given in the simulation, and as can be seen from a tracking curve in fig. 2, the designed fixed-time terminal sliding mode controller works stably under the condition of interference, has good robustness, and can well execute a task of tracking a three-dimensional space track; meanwhile, in order to more intuitively display a good tracking effect, fig. 3 shows the tracking conditions of the aircraft in the x, y and z directions respectively; the tracking effect curves of the three attitude variables theta, psi and phi are shown in FIG. 4; the diagram shows that the roll angle phi can be tracked to a desired value in a short time, and the pitch angle theta and the yaw angle psi are kept stable in the flying process and accord with a desired control effect;
FIG. 5 is a variation curve of position tracking error, the tracking errors in the x-axis, y-axis and z-axis directions can be converged to zero quickly, the adjustment time is about 5s, and is less than the upper limit T of the position convergence timeout8.0255 s; FIG. 6 shows the tracking error curves of three attitude variables, from which it can be clearly seen that the adjustment time of the error convergence is extremely short, about 1s, less than the upper bound T of the inner ring convergence timein3.1944s, further verifying the validity of the control design; fig. 4 and 6 show the control input quantity of the position, the adjustment effect of the 0-5s controller is obvious, after 5s, the tracking error of the position state is converged to zero, the control law curve tends to be stable, and the whole adjustment process has no obvious buffeting phenomenon. The attitude subsystem control curves are shown in fig. 4 and 7, and it can be seen that the controller has a significant 0-1s function, corresponds to the attitude tracking error convergence time, and has a good control effect;
the fixed-time four-rotor aircraft control method based on terminal sliding mode control has the advantages of good convergence speed and tracking capability, rapidity and robustness.
The foregoing shows and describes the general principles, essential features, and advantages of the invention. It will be understood by those skilled in the art that the present invention is not limited to the embodiments described above, which are described in the specification and illustrated only to illustrate the principle of the present invention, but that various changes and modifications may be made therein without departing from the spirit and scope of the present invention, which fall within the scope of the invention as claimed. The scope of the invention is defined by the appended claims and equivalents thereof.
Claims (7)
1. A fixed-time four-rotor aircraft control method based on terminal sliding mode control is characterized by comprising the following steps: comprises that
The method comprises the following steps: firstly, establishing a nonlinear dynamics model of a quadrotor aircraft based on a Lagrange equation;
step two: converting a nonlinear dynamics model of the four-rotor aircraft into a second-order nonlinear system form comprising a position system and an attitude system;
step three: intermediate command signal theta for solving position model based on under-actuated characteristic of four-rotor aircraftd,ψd；
Step four: designing a nonsingular terminal sliding mode function based on a fixed time theory according to the model processing results of the second step and the third step;
step five: and designing a nonsingular terminal sliding mode fixed time controller by taking the nonsingular terminal sliding mode function in the step four as a control strategy, so that the tracking error of the position and posture trajectory of the system is converged to zero in fixed time.
2. The control method of the fixed-time four-rotor aircraft based on the terminal sliding-mode control is characterized by comprising the following steps of: the nonlinear dynamical model of the quadrotor aircraft based on the Lagrange equation in the first step is as follows:
wherein: the euler angles for the three attitudes of the aircraft are denoted as Ω ═ Φ, θ, ψ]Respectively representing a roll angle, a pitch angle and a yaw angle; angular velocity is expressed as
The position coordinate of the center of mass of the aircraft in the inertial coordinate system is represented as P ═ x, y, z](ii) a The velocity is expressed as
Aircraft radius length l representsThe distance of each rotor tip to the center of gravity of the aircraft; m represents the total load weight of the four-rotor aircraft; i isiRepresenting the moment of inertia about each axis; kiIs a coefficient of resistance; di(i ═ 1,2,3,4,5,6) as perturbations, and a time-varying perturbation d is assumedi1.. 6 is bounded and known in the upper bound, i.e. there is a positive real number λ, such that | diThe | is less than or equal to lambda, and all disturbances are bounded;
using F as the thrust generated by each rotor of the aircraftiDenotes uiFor virtual control input, i ═ 1,2,3,4, …, defined as follows:
wherein: r represents a proportionality coefficient.
3. The control method of the fixed-time four-rotor aircraft based on the terminal sliding-mode control is characterized by comprising the following steps of: step two the process of converting the nonlinear dynamics model of the four-rotor aircraft into a second-order nonlinear system form specifically comprises:
s201, firstly, setting the virtual control input to be designed as follows:
the four-rotor dynamics model used to describe the position states in equation (1) becomes:
s202, order up＝[u1x,u1y,u1z]T，fp＝-[0,0,g]T-diag([K1/m,K2/m,K3/m]) V, let P ═ x, y, z]Representing three-dimensional position, v representing velocity, the quad-rotor aircraft in equation (1)The position model can be written in the form of a second order nonlinear system as follows:
s203. in the same way, make uο＝[u2,u3,u4]T，fο＝-diag[lK4/I1,lK5/I2,lK6/I3]·ω，dο＝[d4,d5,d6]TThen, the quad-rotor aircraft attitude model of equation (4) can be written in the form of a second-order nonlinear system as follows:
s204. set
The quad-rotor aircraft system can be converted to the following second-order system form:
4. the control method of the fixed-time four-rotor aircraft based on the terminal sliding-mode control is characterized by comprising the following steps of: step three, the solving process for solving the intermediate command signal based on the under-actuated characteristic of the four-rotor aircraft comprises the following steps:
s301. available from control inputs of a quad-rotor aircraft:
s302. because
Equation (8) becomes:
s303. due to u1z＝u1cosφcosψdIs obtained by
Equation (9) becomes:
s304, the following formula (10) can be obtained:
s305. at this time, psi can be solved according to the formula (11)dAnd thetadComprises the following steps:
5. the fixed-time four-rotor aircraft control method based on terminal sliding mode control according to claim 4, characterized in that: theta in step S305dThe virtual reference instruction is
Wherein:
6. the control method of the fixed-time four-rotor aircraft based on the terminal sliding-mode control is characterized by comprising the following steps of: the specific process for designing the nonsingular terminal sliding mode function based on the fixed time theory comprises the following steps:
s401, aiming at a second-order nonlinear system
If x is 0, the equilibrium state of the system is defined, and if there is a continuous radially unbounded function V: r → R+U {0}, so that
And the arbitrary solution x (t) of the system satisfies the formula
In formula (14): a. b, p, q and k are positive numbers and satisfy pk < 1 and qk > 1, then the zero balance state of the system is globally fixed time stable, and the solution time upper limit T satisfies the following inequality:
s402, setting a tracking error
Constructing the following nonsingular terminal sliding mode surfaces according to a fixed time theory:
in the formula (16), ai＞0，bi＞0，pj i,qj iAre all positive odd numbers, j is 1,2,3, …, and have
7. The control method of the fixed-time four-rotor aircraft based on the terminal sliding-mode control is characterized by comprising the following steps of: the design process of the nonsingular terminal sliding mode fixed time controller comprises the following steps:
in formula (17), k is 2 λ, and the nonlinear function μiThe definition is as follows:
in the formula (18), when x → 0, the nonlinear function μi(x)/x→0。
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